Nonlinear Approximation Theory

290 Pages·1986·8.07 MB·English

by  Dr. Dietrich Braess (auth.)

Spri nger Series in
Computational
Mathematics
7
Editorial Board
R. L. Graham, Murray Hill
J. Stoer, Wurzburg
R. Varga, Cleveland
Dietrich Braess
Nonlinear
Approximation
Theory
With 38 Figures
Springer-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Profmor DL DiBtrich Braess
Fakultat fUr Mathematik. Ruhr-Universitat Bochum
Postfach 102148.0-4630 Bochum 1
Mathematics Subject Classification (1980):
41-02. 41A15. 41A20. 41A21. 41A25. 41A30. 41ASO. 41A52. 41A55.
41A65. 65010. 65015. 65032
ISBN-131:978-3-642-64883-0 e-ISBN-13: 978-3-642-61609-9
001: 10.1007/978-3-642-61609-9
Library of Congress Cataloging-In-Publication Data
Braess, Deitrich. 1938-
Nonlinear approximation theory.
(Springer series in computational mathematics; 7)
Bibliography: p. Includes index.
1. Approximation theory. I. TItle. II. Series.
OA221.B67 1986 511'.4 86-10101
ISBN-131:978-3-642-;648!l3-0
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© Springer-Verlag Berlin Heidelberg 1986
Typesetting: Asco Trade Typesetting Ltd., Hong Kong
Printing and bookbinding: Graphlscher Betrieb Konrad Triltsch, WUrzburg
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To Anneliese
Preface
The first investigations of nonlinear approximation problems were made by
P.L. Chebyshev in the last century, and the entire theory of uniform approxima
tion is strongly connected with his name. By making use of his ideas, the theories
of best uniform approximation by rational functions and by polynomials were
developed over the years in an almost unified framework. The difference between
linear and rational approximation and its implications first became apparent in
the 1960's.
At roughly the same time other approaches to nonlinear approximation were
also developed. The use of new tools, such as nonlinear functional analysis and
topological methods, showed that linearization is not sufficient for a complete
treatment of nonlinear families. In particular, the application of global analysis
and the consideration of flows on the family of approximating functions intro
duced ideas which were previously unknown in approximation theory. These
were and still are important in many branches of analysis.
On the other hand, methods developed for nonlinear approximation prob
lems can often be successfully applied to problems which belong to or arise from
linear approximation. An important example is the solution of moment problems
via rational approximation. Best quadrature formulae or the search for best
linear spaces often leads to the consideration of spline functions with free nodes.
The most famous problem of this kind, namely best interpolation by poly
nomials, is treated in the appendix of this book.
The monograph grew out of lectures which the author gave on numerous
occasions to fourth year students at the Ruhr-UniversiHit Bochum. The pre
requisites consist essentially of a good basic knowledge of analysis and functional
analysis. A short description of an elementary part of nonlinear approximation
suitable for a course for post-graduate students is given in the "Note to Students".
The note indicates sections to which a student may restrict himself during a first
reading.
It is hoped that this book will prove to be useful for researchers interested in
advanced aspects of approximation theory. In this respect the book has been
organized so that the discussions of rational functions (Chap. V), exponential
sums (Chap. VI and VII), and spline functions with free nodes (Chap. VIII) is in
each case independent. We assume only that the reader is accustomed to the
basic methods from functional analysis (described in Chap. II, § 1) and the central
ideas of critical point theory (Chap. III). The latter parts of Chap. VII and Chap.
viii Preface
VIII require topological arguments and techniques from global analysis which
are not elementary. These are presented in Chap. IV.
We have deliberately abandoned a unified theory for the special families
described above, since it would conceal more facts than it would make trans
parent. Usually, there are three ingredients in the proofs: 1) a local argument
from differential calculus, 2) a global argument which often involves a topological
argument, 3) a compactness argument. Frequently, the compactness part is the
most strongly involved. In the families under consideration only weak compact
ness conditions are fulfilled. The methods for overcoming this difficulty depend
on the particular nature of the special case. The exercises indicate which results
for a particular family carryover to the others. Moreover, the exercises contain
some curiosities and interesting examples or counterexamples which could not
be given in the text.
In attributing proper names to the theorems, we have generally followed the
common usage in the western literature. The bibliographical notes should not
be considered as being complete. We hope for example, that the reader will excuse
the many gaps in the Russian literature apparent in our bibliography.
I wish to thank most of all Helmut Werner, to whom I personally owe so
much. He created a very stimulating scientific atmosphere at the Institute fUr
Numerische Mathematik ofthe University of Munster, and this not only because
of his deep knowledge of nonlinear analysis. I express my sincere gratitude to
Ward Cheney for inviting me to participate in discussions and seminars at the
University of Texas at Austin. These were important for the development of
critical point theory as it is presented here. A number of my friends and colleagues
were kind enough to read large parts of the manuscript and made improvements
and valuable suggestions. For this I am especially indebted to Frank Deutsch,
Hubert Jongen, Allan Pinkus, Robert Schaback, JosefStoer, Richard Varga, Luc
Wuytack, and A. Zhensykbaev. I also wish to thank Timothy Norfolk for
linguistic improvements. For their expert and diligent typing and retyping of the
manuscript, I thank Mrs. I. Voigt and Mrs. M. Schulz. I am grateful to Ludwig
Cromme, Immo Diener, Kurt Jetter, Manfred Muller, and Robert Schaback for
their help in proofreading. Finally, I would like to thank Springer-Verlag for
their friendly cooperation.
Bochum, Spring 1986 Dietrich Braess
Note to Students
Large parts of the text are based on lectures which the author gave to fourth
year students at German universities. Though this monograph is directed to
researchers interested in approximation theory, it is also our intention to provide
a text for students. In this respect, the book was designed so that the student
may restrict himself to the basic parts of nonlinear approximation theory. The
following hints are intended to explain which sections may be skipped.
The first chapter should be regarded as a review of well-known results from
the linear theory with which the reader should be acquainted. For later applica
tions it is convenient to consider this in the framework of convex approximation.
The concept of strong uniqueness, which is described in the framework of
Chebyshev approximation will turn out to be essential. On the other hand, the
characterization of nearest points in Haar cones in § 3C and the Lt-theory in § 4
will only be used in some special situations.
The theorem that each boundedly compact Chebyshev set in a smooth
Banach space must be convex, is a main result of Chapter II. To understand
this one should look at the general tools which are useful for existence proofs
(Section 1A), proceed with the discussion on suns and the Kolmogorov Criterion
in § 2 until Theorem 2.5, continue in § 3A until Corollary 3.2, and in § 3B until
Corollary 3.6.
Methods of local analysis are described in Chapter III. The first section is
crucial for the understanding of the rest of the book, but the student may restrict
himself to manifolds without boundaries and skip all generalizations which refer
to boundaries. Next, the basic facts on varisolvent families from sections 3A and
3B will often be applied in nonlinear Chebyshev approximation. Additional tools
and improvements for the differentiable case are provided in 4A-C.
Chapter IV is intended to give some introduction to the use of global
methods. In § 1 where some basic concepts are briefly discussed, it is sufficient to
understand the definition of critical points of mountain pass type. In lectures the
author has only presented either § 2 or § 3. This is enough to give the student
some insight into global analysis. For this purpose, the student may choose
between the general uniqueness theorem in §2A-B and a concrete example
in § 3.
The basic facts on rational approximation in Chapter V, §§ 1-2, familiarize
the reader with the most frequently studied nonlinear family which behaves
"almost" like a linear one.
x Note to Students
The first contacts with Descartes' rule of signs are encountered in the analysis
of exponential sums in Chapter VI, § 1. The theory in § 2 shows that existence
proofs in nonlinear problems may require hard analysis and that general existence
theorems may support, but cannot replace the individual proofs.
The y-polynomials in Chapter VII are generalizations of exponential sums.
Thus the student may skip § 1 by taking Descartes' rule for exponential sums
from Chapter VI. The basic results are developed in §§ 2 and 3.
The investigations in Chapter VIII are only aimed at those who are partic
ularly interested in spline functions or mono splines and Gaussian quadrature
formulas. Although the analysis up to Section 3B is not difficult, one always has
the added technical complication caused by the interlacing condition.
The appendix on optimal Lagrangian interpolation and the non-elementary
sections of Chapters VII and VIII make extensive use of advanced methods of
global analysis.
Finally, we mention that the book contains two small complete theories
which are suitable for seminars.
A seminar for students who are interested in functional analysis may be based
on Chapter II, augmented by Section 1.2A for an introduction.
A seminar concerned with the connection between rational approximation
and classical constructive function theory could focus on Chapter V. The only
prerequisites here are Lemma 11.1.4, the lemma of first variation in connection
with Exercise 111.1.23, and the well known theorem of de la Vallee-Po us sin.
Contents
Chapter I. Preliminaries........................................ 1
§ 1. Some Notation, Definitions and Basic Facts. . . . . . . . . . . . . . . . . . . . 1
A. Functional Analytic Notation and Terminology. . . . . . . . . . . . . . 1
B. The Approximation Problem. Definitions and Basic Facts . . . . . 3
C. An Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
D. Divided Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§ 2. A Review of the Characterization of Nearest Points in Linear and
Convex Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A. Characterization via the Hahn-Banach Theorem and the
Kolmogorov Criterion . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
B. Special Function Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
§ 3. Linear and Convex Chebyshev Approximation. . . . . . . . . . . . . . . . . . 11
A. Haar's Uniqueness Theorem. Alternants. . . . . . . . . . . . . . . . . . . . 12
B. Haar Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
C. Alternation Theorem for Haar Cones. . . . . . . . . . . . . . . . . . . . . . . 16
§4. L1-Approximation and Gaussian Quadrature Formulas. . . . . . . . . . 18
A. The Hobby-Rice Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
B. Existence of Generalized Gaussian Quadrature Formulas...... 19
C. Extremal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter II. Nonlinear Approximation: The Functional Analytic
Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
§ 1. Approximative Properties of Arbitrary Sets. . . . . . . . . . . . . . . . . . . . . 24
A. Existence............................................... 24
B. Uniqueness from the Generic Viewpoint. . . . . . . . . . . . . . . . . . . . 27
§ 2. Solar Properties of Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
A. Suns. The Kolmogorov Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 31
B. The Convexity of Suns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
C. Suns and Moons in C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
§ 3. Properties of Chebyshev Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A. Approximative Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
B. Convexity and Solarity of Chebyshev Sets. . . . . . . . . . . . . . . . . . . 40
C. An Alternative Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xii Contents
Chapter III. Methods of Local Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
§ 1. Critical Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A. Tangent Cones and Critical Points. . . . . . . . . . . . . . . . . . . . . . . . . 48
B. Parametrizations and C1-Manifolds. . . . . . . . . . . . . . . . . . . . . . . . 51
C. Local Strong Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
§2. Nonlinear Approximation in Hilbert Spaces. . . . . . . . . . . . . . . . . . . . 58
A. Nonlinear Approximation in Smooth Banach Spaces. . . . . . . . . . 58
B. A Classification of Critical Points. . . . . . . . . . . . . . . . . . . . . . . . . . 59
C. Continuity............................................. 61
D. Functions with Many Local Best Approximations. . . . . . . . . . . . 63
§ 3. Varisolvency............................................... 66
A. Varisolvent Families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
B. Characterization and Uniqueness of Best Approximations. . . . . 68
C. Regular and Singular Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
D. The Density Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
§4. Nonlinear Chebyshev Approximation: The Differentiable Case. . . . 75
A. The Local Kolmogorov Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . 76
B. The Local Haar Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
C. Haar Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
D. The Local Uniqueness Theorem for C1-Manifolds. . . . . . . . . . . . 79
§ 5. The Gauss-Newton Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
A. General Convergence Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B. Numerical Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter IV. Methods of Global Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 88
§ 1. Preliminaries. Basic Ideas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A. Concepts for the Classification of Critical Points . . . . . . . . . . . . . 88
B. An Example with Many Critical Points. . . . . . . . . . . . . . . . . . . . . 90
C. Local Homeomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
§ 2. The Uniqueness Theorem for Haar Manifolds. . . . . . . . . . . . . . . . . . . 93
A. The Deformation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B. The Mountain Pass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
C. Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
§ 3. An Example with One Nonlinear Parameter. . . . . . . . . . . . . . . . . . . . 100
A. The Manifold E~\E~_l ................................... 101
B. Reduction to One Parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C. Improvement of the Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Chapter V. Rational Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
§ 1. Existence of Best Rational Approximations. . . . . . . . . . . . . . . . . . . . . 108
A. The Existence Problem in C(X). . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
B. Rational Lp-Approximation. Degeneracy. . . . . . . . . . . . . . . . . . . . 110
§ 2. Chebyshev Approximation by Rational Functions. . . . . . . . . . . . . . . 113